Finding Homogeneous Solutions for a Second Order Differential Equation

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SUMMARY

The discussion focuses on finding homogeneous solutions for the second-order differential equation y'' - 4y' + 4y = x * e^(2x). The characteristic equation derived from the homogeneous part, y'' - 4y' + 4y = 0, leads to a double root at a = 2, resulting in one solution C1 * e^(2x). The second solution, due to the presence of the double root, is C2 * x * e^(2x), confirming that both solutions are valid for the homogeneous equation.

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  • Understanding of second-order differential equations
  • Familiarity with characteristic equations
  • Knowledge of homogeneous solutions and their properties
  • Basic skills in solving differential equations
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  • Study the method of undetermined coefficients for non-homogeneous equations
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Students studying differential equations, mathematics educators, and anyone involved in applied mathematics or engineering looking to deepen their understanding of homogeneous solutions.

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Homework Statement


y''-4y'+4y=x*e2x

I'm trying to find the homogenous solutions of this equation. I know there are two, but I can only find one.

YH=> y''-4y'+4y=0

Homework Equations





The Attempt at a Solution



y''-4y'+4y=0
Using the characteristic function:
a2-4a+4=a <=> (a-2)2=0
Therefore
C1*e2x
is a solution. However, since this is a 2nd order differential equation, I should have two. I can solve the rest of the equation if I can find the 2nd solution. I've tought about it a lot but can't manage to find the answer... All help will be appreciated.
Thanks!
 
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If you have a double root 2, then C2*x*e^(2x) is also a homogeneous solution.
 
Last edited:

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